## Inequalities

Inequalities

Inequalities are mathematical sentences containing signs such as:

 Sign Meaning Example < is less than 3 < 4 > is greater than 4 > 3 ≤ is less than or equal to y ≤ 5 ≥ is greater than or equal to x ≥ 3

Inequations, or inequalities, can be solved in the same way as equations.

Example

2q + 3 > 4

2q > 1

q > 0.5

The only difference between equations and inequations is that when both sides of the inequation are

multiplied or divided by a negative number, the inequality sign must be reversed.  ### Word Problems involving Inequalities

As with equations, some word problems can be solved by forming an inequality.

e.g. The sum of two numbers is less than 10 can be written mathematically as x + y < 10

Example

A man has \$700 to spend on a suit and some shirts. He finds a new suit for \$450 and the shirts cost \$45 each. How many shirts can he buy?

Step 1 Form an inequality

Let the number of shirts the man can buy be x

45x + 450 ≤ 700

Step 2 Solve the inequality

45x + 450 ≤  700
45x ≤  700 − 450
45x ≤  250
x ≤  5.6 (to 2 s.f.)

Step 3 Write out the solution to the problem in words.

If x ≤  5.6 then he can buy 5 shirts

### Number Lines and Inequalities

When an inequality has been solved the solution can be shown on a number line.

e.g. Solve 3x + 6 < 21 for x is a member of R (x is a real number)

3x <15

x < 5 